Coefficient bounds for Banach-valued nonvanishing holomorphic maps

Determine sharp bounds for the norms of the Taylor coefficient vectors c_n in the expansion f(z) = c_0 + c_1 z + c_2 z^2 + ⋯ for nonvanishing holomorphic maps f: D → X whose values lie in the unit ball B(X) of a complex Banach space X. This problem extends the Hummel–Scheinberg–Zalcman coefficient problem from scalar Hardy spaces to Banach-valued holomorphic functions and seeks explicit, optimal estimates on the coefficient norms under the nonvanishing and bounded-range constraints.

Background

Section 9 proposes extending the classical Hummel–Scheinberg–Zalcman (HSZ) coefficient problem—originally for scalar nonvanishing H^p functions on the unit disk—to the setting of holomorphic maps taking values in a complex Banach space. The paper introduces B(X) as the unit ball of X and discusses two special cases where restricted solutions are available, indicating that the general Banach-valued problem remains unresolved.

This extension is motivated by the desire to generalize scalar coefficient bounds (e.g., HSZ and Krzyz-type estimates) to vector-valued holomorphic functions, where coefficients become elements of X. The author highlights the lack of general bounds and frames the task of deriving sharp coefficient estimates under nonvanishing and bounded-range hypotheses.